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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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New subclasses of the class of close-to-convex functions
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by Pran Nath Chichra PDF
Proc. Amer. Math. Soc. 62 (1977), 37-43 Request permission

Abstract:

In this paper we introduce new subclasses of the class of close-to-convex functions. We call a regular function $f(z)$ an alpha-close-to-convex function if $(f(z)f’(z)/z) \ne 0$ for z in E and if for some nonnegative real number $\alpha$ there exists a starlike function $\phi (z) = z + \cdots$ such that \[ \operatorname {Re} \;\left [ {(1 - \alpha )\frac {{zf’(z)}}{{\phi (z)}} + \alpha \frac {{(zf’(z))’}}{{\phi ’(z)}}} \right ] > 0\] for z in E. We have proved that all alpha-close-to-convex functions are close-to-convex and have obtained a few coefficient inequalities for $\alpha$-close-to-convex functions and an integral formula for constructing these functions. Let ${\mathfrak {F}_\alpha }$ be the class of regular and normalised functions $f(z)$ which satisfy $\operatorname {Re} \;(f’(z) + \alpha zf''(z)) > 0$ for z in E. $f(z) \in {\mathfrak {F}_\alpha }$ gives $\operatorname {Re} f’(z) > 0$ for z in E provided $\operatorname {Re} \alpha \geqslant 0$. A sharp radius of univalence of the class of functions $f(z)$ for which $zf’(z) \in {\mathfrak {F}_\alpha }$ has also been obtained.
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Additional Information
  • © Copyright 1977 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 62 (1977), 37-43
  • MSC: Primary 30A32
  • DOI: https://doi.org/10.1090/S0002-9939-1977-0425097-1
  • MathSciNet review: 0425097